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Monday, 27 November 2017

Although there was no Dev Patel, or tigers in this pi, we still managed to squeeze a boat into this week's puzzle.

This puzzle is a variant on an old favourite. We don't have chickens, corn, or foxes, but we still want to cross a river. Here's the question:

A husband and wife come to the edge of a river, where they find two children with a small boat. The boat can hold either one child, two children, or one adult. How do you get everyone across the river in the minimum possible number of crossings?

Post your answers below, and we'll give you the answer in the next podcast!

Thursday, 16 November 2017

I have two children. One of them is a boy and they were born on a Tuesday.

What is the probability that both children are boys?

This is a hard question, and Ben ****ed up the explanation when he tried to do it live. So, as penance, we made him sit down and explain it as a video.

Here's a simpler question written out much nicer:

I have two children. One of them is a boy.
What is the probability that both children are boys?

Now you may think the probability is 50%, but that is not so (note that we are assuming that boy and girl births are equally likely). The reason is because we have more information about the children.

Suppose we denote a boy by "b" and a girl by "g". Further, we capitalise the letter to denote the elder child. In this way we could have the following combinations of children:

Bb

Gb

Bg

Gg

However, we know we have at least one boy, so we can't have Gg. Out of the possibilities that are left, namely Bb, Gb and Bg, there is only one way to get two boys, the chance is 1/3! Counter-intuitive no?

Note that if we had posed the problem as I have two children and my eldest is a boy then (using the above argument) the probability of have a second boy is then 1/2.

Probability can be a tricksy animal. Even for a Cambridge educated lecturer!

Wednesday, 15 November 2017

A simple one to start you off.
I buy a bottle and a cork for £1.10. The bottle costs £1 more than the cork.

How much does the cork cost?

A moments thought should show that the bottle costs £1.05, whilst the cork costs only 5p. If you go it right first time well done! The answer most people tend to give if they don't pause for a second is 10p.

Now, for the more difficult question:

I
live on a street with more than one house. All the houses on this
street are numbered consecutively, 1, 2, 3,..., etc. Amazingly, I live
in the house such that if you add up all the house numbers below me and
all the house numbers above me then they come to exactly the same
answer.

What is the minimum number of houses on this street and what is my house number?

The smallest answer, excluding the one house case is 8 houses on the street and I live at number 6, thus, 1+2+3+4+5=15=7+8.

There are actually an infinite number of increasing solutions to this problem. Although the solution can be found using basic algebra and a knowledge of continued fractions the details can get a bit hairy. Thus, I direct the interested reader to the following two wonderful expositions on the matter:

Tuesday, 14 November 2017

Ben's greengrocer uses a balance scale, like the one seen above, and only has a 40kg weight. However, the greengrocer fortuitously broke the weight into four pieces of integer weight that will allow them to measure out
every integer of kilograms from 1kg to 40kg. What are the four weights?

Normally, for a problem like this, you'd think of the binary sequence 1, 2, 4, ..., because, as shown in the gold chain problem, you can construct any number using combinations of these numbers. However, with only four weights, we would have 1, 2, 4, 8, from which we could produce a maximum of 15, falling far short of the 40kg total.

The crux of the problem is that in binary we can only add or not add a weight. In this problem, because we are using a set of balance scales we have three possibilities:
(a) Not add the weight, denoted 0;
(b) Add the weight to the left side, denoted L;
(c) Add the weight to the right side, dented R.

Because we have three possibilities, instead of two, we might think about using the numbers based around powers of 3 (the trinary system), rather than those based around powers of 2 (the binary system). Thus, our weights would be 1, 3, 9, 27. Adding these together does indeed give 40kg, but how would we use them to weigh out 2kg?

Put the 1kg on the left and the 3kg on the right. This produces a deficit of 2kg in the left pan, so we add apples to the left pan until it balances and, voila, we know we have two kilograms of apples.
Thus, 2 is represented as LR00 in our system.

What about 5kg? Similar to the above put the 1 and 3kg weights in the left and the 9 in the right. This produces a deficit of 5kgs in the left pan. Thus, 5kg is represented by LLR0.

Friday, 10 November 2017

I have two children. One of them is a boy and they were born on a Tuesday.

What is the probability that both children are boys?

Now, unfortunately, Ben royally screws up the explanation of this answer in the next podcast. However, we are recording an appendix episode to put the world to rights. In the mean time though we can solve the slightly simpler puzzle:

Thursday, 9 November 2017

A simple one to start you off.
I buy a bottle and a cork for £1.10. The bottle costs £1 more than the cork.

How much does the cork cost?

And a more difficult one for you to chew on that was apparently given to Ramanujan himself.

I live on a street with more than one house. All the houses on this street are numbered consecutively, 1, 2, 3,..., etc. Amazingly, I live in the house such that if you add up all the house numbers below me and all the house numbers above me then they come to exactly the same answer.

What is the minimum number of houses on this street and what is my house number?

Wednesday, 8 November 2017

Until recently my greengrocer sold apples in multiples of 40kg. The apples are weighed out using an old set of balance scales and a 40kg weight (see the above picture). However, the greengrocer dropped their weight and it broke into four pieces weighing four different integer values. Just before the greengrocer threw the pieces away I stopped them and showed that the set of four smaller weights could be used to measure out every integer of kilograms from 1kg to 40kg.

Friday, 3 November 2017

Alongside your regular team of Thomas, Ben and Liz there was only one mathematician with the expertise who could take us through this movie with grace, wit and wisdom. And that mathematician wasn't available so we got

Dr James Grimes

instead.

Join us for episode five of Maths at: The Movies as we separate fact from fiction about the life of Alan Turing.